Various centroids and some characterizations of catenary rotation hypersurfaces

Authors:
DONG-SOO KIM, YOUNG HO KIM, DAE WON YOON

Abstract:
We study positive $C^1$ functions $z=f(x), x=(x_1,\cdots, x_n)$ defined on the $n$-dimensional Euclidean space $ \mathbb R^{n}$.
For $x=(x_1,\cdots, x_n)$ with nonzero numbers $x_1, \cdots, x_n$, we consider the rectangular domain
$I(x)=I(x_1)\times \cdots \times I(x_n)\subset \mathbb R^{n}$,
where $I(x_i)= [0, x_i]$ if $x_i>0$ and $I(x_i)= [x_i,0]$ if $x_i<0$.
We denote by $V$, $S$,
$(\bar{x}_{V},\bar{z}_{V})$,
and
$(\bar{x}_{S},\bar{z}_{S})$
the volume of the domain under the graph of $z=f(x)$, the
surface area $S$ of the graph of $z=f(x)$,
the geometric centroid of the domain under the graph of $z=f(x)$,
and the surface centroid of the graph itself over the rectangular domain $I(x)$, respectively. In this paper, first we show that among
$C^2$ functions with isolated singularities, $S=kV$, $k\in \mathbb R$
characterizes the family of catenary rotation hypersurfaces
$f(x)=k\cosh(r/k)$, $r=|x|$.
Next we show that the equality of $n$ coordinates of
$(\bar{x}_{S},\bar{z}_{S})$ and $(\bar{x}_{V},2\bar{z}_{V})$ for every rectangular domain
$I(x)$ characterizes the family of catenary rotation hypersurfaces among
$C^2$ functions with isolated singularities.